HMMT 2021 G9

Let $A B C D$ be a trapezoid with $A B \| C D$ and $A D=B D$. Let $M$ be the midpoint of $A B$, and let $P \neq C$ be the second intersection of the circumcircle of $\triangle B C D$ and the diagonal $A C$. Suppose that $B C=27, C D=25$, and $A P=10$. If $M P=\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$.